In Civ V, can I improve a resource that is surrounded on all 6 sides by mountains?

Solution 1:

You're out of luck. There's no way to get past the mountain range with workers.

EDIT

Actually, if legions upgraded to paratroopers, and if you use e.g. the balance mod, which connects resources if there's a fort, you could connect the resource. Alas, legions are not on the right upgrade path, so you're still screwed.

Solution 2:

If another civilization has a border close to the mountain range it might be possible to teleport in by declaring war with your worker positioned close next to the mountain in soon to be enemy territory. When the worker is evicted I believe it will move to the closest valid tile.

Not very practical though.